We show the interest of nonparametric methods taking into account the boundary correction techniques for a numerical evaluation of an approximation error between the stationary distributions of
When modeling practical problems, one may often replace a real system by another one which is close to it in some sense but simpler in structure and/or components. This approximation is necessary because real systems are generally very complicated, so their analysis cannot lead to analytical results or it leads to complicated results which are not useful in practice.
To overcome the difficulties encountered in obtaining exact and interpretable solutions for many queueing systems, analysts use approximation methods. Use of these methods allows approaching the characteristics of a complex model by those of a simpler one. It is interesting in this case to measure the resulting approximation error.
One of these approximation methods is the strong stability [
In this paper, we focus on the evaluation of the approximation error between the stationary distributions of
Moreover, as the strong stability method assumes that the perturbation is small, then we suppose that the arrivals law of the
On the other hand, in practice, we are often more interested in the deviation between the average characteristics (e.g., the mean waiting time) of the ideal model and the perturbed one than in the difference between stationary probabilities. Indeed, in most of the cases the
This paper is organized as follows: in Section
To determine the proximity of the stationary distributions
In this section, we introduce some necessary notations appropriate for our case study and recall the basic definition of the strong stability method. For a general framework, see [
Consider the measurable space
Let
Let us consider
Let
A Markov chain
Let
Suppose that the traffic intensity
The margin between the transition operators is given by
In addition, if
Given the error
Let
In practice, the critical step in the kernel density estimation is the choice of the bandwidth
Several results are known in the literature when the density function is defined on the real line
Schuster [
Recall that our object is the numerical evaluation of the approximation error between the stationary distributions of
(1)
(2) Introduce the sample size
(3) Determine the solution
(4)
(5) Computation of the proximity error:
(6)
(0)
(1) Generate a sample of size
theoretical density
(2) Estimate the theoretical density
in general
(3) Introduce the mean service rate
(4) Determine the mean arrival rate
(5) Verify the stability: if
else
(6) Determine the proximity of
(7) Determine the approximation domain (
if
else
(8) Determine the minimal error
(9)
See Algorithms
We consider the following four cases.
For the last three cases, we take the sample size
We first determine the interarrival mean rate
interarrival mean rate:
traffic intensity:
For calculation of the proximity error of the
Proximity error







0.1475  0.0668  0.1223  0.1364 

0.0142  0.0665  0.0220  0.0170 
In most of the cases the
Using results of Table
Note that according to Figure
Comparison of waiting times distributions of
To realize this work, we use Algorithm
We generate samples of size
Exp(1)  Weibull(2, 0.5, 0)  Gamma(1, 3)  





Interarrival mean time 
0.9190  1.8244  0.2750 
Traffic intensity 



Stability domain 



Variation 
0.2444  0.3502  0.1615 
Note also that according to Table
Following the previous example, the classical kernel estimate (ParzenRosenblatt estimate) has shown its insufficiency for determining the approximation error on the stationary distributions of the corresponding systems. It is why we consider again in this example the study of this problem by using the same hyperexponential law defined in (






Variation distance 
0.0711  0.2104  0.0895  0.0792 
Error Err  0.21  0.35  0.26 
Error
Notice following Figure
In practice, we are often more interested in the deviation between the average characteristics (e.g., the mean queue length) of the nominal chain and the perturbed one than in the difference between stationary probabilities. For this purpose, we give Corollary
Suppose that the assumptions of Theorem
The proof follows from Theorem
Let
By dividing expression (
Applying this formula and results of Table
Error on mean waiting times of





1.3  1.31 
Err  0.21  0.26 

0.1699  0.2028 
By comparing the results of Tables
Again, following the results of Figure
To summarize, the comparative study between the results obtained by applying the different nonparametric methods to both considered norms and for some specific characteristics shows the impact and interest of those that take into account the correction of boundary effects to determine an approximation error between the considered systems (
Systems used in this paper are relatively simple. They serve more as an illustrative support for a good comprehension of the techniques used to solve the posed problem. It would be interesting to consider the results of this work for the approximation of more complex systems, such as the
The authors declare that there is no conflict of interests regarding the publication of this paper.